My brother has a J.Stevens & Co. pocket rifle that has a patent date of Sept. 6, 1864. It has an 18" smooth bore barrel and a metal screw on stock and a pistol grip. It was made after 1864 but before 1886 when the company name was changed. The problem we have is we don't know what caliber it is. A 44 cartridge is to small and a 45 is to large. What could it be? What is a 44-40?

I think you need to figure out first if it is rimfire or centerfire by looking at the firng pin. Figure out your bore diameter, Then you need to look at cartridges of the world. There are lots of different cartridges from the 1850 to 1900 time period that were basically dead ends like the Stevens 22 Extra Long and many more.

Your brother doesn't have a Stevens Pocket Rifle (it's a smoothbore) - he has a Stevens No.42 Pocket Shotgun, most likely chambered for the special .44 Everlasting shells (the .44-40 was an option, since it was available in a shot loading before the .410 came along).
They were available as either a 15" bbl ($11) or an 18" bbl ($12) model at the time.

The Everlasting cartridges were just a heavy walled cartridge that could be reloaded almost indifiently.
Almost any Ballard or Stevens rifle could be ordered chambered for Everlasting cases, but, in most cases could also fire normal factory thin walled cartridges.
The .44/50 and .44/65 Stevens Everlasting cartridges were the shot shells primarily used in the Stevens New Model Pocket Shot Gun and in the Stevens Hunters Pet Shot Gun. (Both of these models were smooth bored.)

I wouldn't recommend he (or anyone) fire any .410 ammo in it, by any means.

A .410 is just slightly larger. The brass will go in about half way. Are the 44.40 and 44 everlasting the same size? He took it in trade about 25 years ago in exchange for doing some bodywork on a car. It was traded to him as a 45/410 but he didn't think about it until this week when I decided to clean it when I was cleaning my weapons (it's been at my house for years). Photo attachment here:

It is as solid and tight as a new gun. I was apparently used very little. We have decided it I a 44-40 shot that was discontinued years ago (almost a .410 but before it-the shell is slightly tapered). We found one shell on gunbroker up for bid with a starting bid of $6 + shipping and a box of 40 that had a starting bid of $495 + $30 shipping.

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